Some aspects of goal-oriented a posteriori error estimation are addressed in the context of steady convection–diffusion equations. The difference between the exact and approximate values of a linear target functional is expressed in terms of integrals that depend on the solutions to the primal...

In order to have reliable numerical simulations it is very important to preserve basic qualitative properties of solutions of mathematical models by computed approximations. For scalar second-order elliptic equations, one of such properties is the maximum principle. In our work, we give a short...

In this paper, we introduce a multi-physics modular hp-FEM system HERMES. The code is based on a novel approach where the finite element technology (mesh processing and adaptation, numerical quadrature, assembling and solution of the discrete problems, a-posteriori error estimation, etc.) is...

We propose a new class of hierarchic higher-order finite elements suitable for the hp-FEM discretization of symmetric linear elliptic problems. These elements use shape functions which are partially orthonormal on the reference domain under the energetic inner product induced by the elliptic...

Most results related to discrete nonnegativity conservation principles (DNCP) for elliptic problems are limited to finite differences and lowest-order finite element methods (FEM). In this paper we show that a straightforward extension to higher-order finite element methods (hp-FEM) in the...

The paper is devoted to complementary approaches in a posteriori error estimation for a diffusion-reaction model problem. These approaches provide sharp and guaranteed upper bounds for the energy norm of the error and they are independent from the way how the approximate solution is obtained. In...